Problem 42 - Advanced Engineering Mathematics

Question

Suppose that \(y_{1}, y_{2}, \ldots, y_{k}\) are \(k\) nontrivial solutions of a homogeneous linear \(n\) th-order differential equation with constant coefficients and that \(k=n+1\). Is the set of solutions \(y_{1}, y_{2}, \ldots, y_{k}\) linearly dependent or linearly independent on \((-\infty, \infty) ?\) Discuss.

Step-by-Step Solution

Answer

The solutions are linearly dependent as they exceed the dimension of the solution space.

Step 1: Identify the equation type

The problem involves a homogeneous linear differential equation with constant coefficients. These kinds of equations typically take the form: \( a_n y^{(n)} + a_{n-1} y^{(n-1)} + \cdots + a_1 y' + a_0 y = 0 \), where the coefficients \(a_i\) are constants.

Step 2: Understand the dependency condition

The solutions of a homogeneous linear \(n\)th-order differential equation form a vector space of dimension \(n\). For \(k=n+1\), the set \(\{y_1, y_2, \ldots, y_k\}\) consists of one more solution than the dimension of the space.

Step 3: Apply linear dependence theorem

By the Linear Dependence Theorem, any set of \(n+1\) vectors in a \(n\)-dimensional space is linearly dependent. This applies to the solutions \(y_1, y_2, \ldots, y_{n+1}\) because they inhabit a function space of dimension \(n\).

Step 4: Conclusion

Since \(k = n+1\), the solutions \(y_1, y_2, \ldots, y_{k}\) must be linearly dependent. They satisfy the dependency condition having more functions than the basis dimension of the solution space.

Key Concepts

Homogeneous Differential Equations

In mathematics, a differential equation is termed **homogeneous** if it satisfies the condition of not containing any term that is a function of the independent variable alone. This means all terms depend on the unknown function and its derivatives. The general form of a homogeneous linear differential equation is given by:\[a_n y^{(n)} + a_{n-1} y^{(n-1)} + \cdots + a_1 y' + a_0 y = 0\]where the function \(y\) and its derivatives appear in each term. Another key feature of these equations is that they are central in determining the behavior of physical systems that are linear.

• Focus: Ensures understanding of how all terms involve the unknown variable \(y\).
• Application: Homogeneous equations are crucial for modeling scenarios in physics, engineering, and other sciences.

To grasp this completely, remember that every solution of this equation can be written as a linear combination of a basic set of solutions. This is crucial when analyzing the complete solution set.

Constant Coefficients

The term **constant coefficients** in differential equations indicates that the coefficients \(a_n, a_{n-1}, \ldots, a_0\) are fixed numbers, not depending on the independent variable. This simplifies the solving process significantly because methods like the characteristic equation can be applied directly without transformation.

• Simplification: The constancy of coefficients makes it easier to predict the behavior of solutions.
• Characteristic Equation: Used to find solutions in terms of exponential functions.

Because the coefficients don't change, the same equations and solutions can often apply to different initial situations, which makes them extremely powerful tools for science and engineering applications.

Linear Independence

**Linear independence** is a concept that determines if a set of functions (or vectors) are independent or dependent on each other. A set of functions \(y_1, y_2, \ldots, y_k\) is said to be linearly independent if no function in the set can be expressed as a linear combination of the others. Mathematically, this means:\[c_1 y_1 + c_2 y_2 + \cdots + c_k y_k = 0\]implies that all coefficients \(c_1, c_2, \ldots, c_k\) must be zero.

• Key Idea: Linear independence reinforces the uniqueness of each solution, ensuring diversity in the solution space.
• Application: In differential equations, having a linearly independent set of solutions allows for the construction of the general solution.

When you find more functions present than the dimension of the space, like in the problem where \(k=n+1\), linear dependence must occur. This is crucial for understanding how solutions to differential equations distribute over their respective space.

Solution Space

The **solution space** of a differential equation is the set of all possible solutions that satisfy the equation. For a linear homogeneous differential equation of order \(n\), the solution space is a vector space. This means it has both the additive property and is closed under scalar multiplication.

• Vector Space Properties: Solutions can be added together, and any solution can be multiplied by a constant.
• Dimensionality: For an \(n\)-th order equation, the solution space has a dimension of \(n\).

Understanding the dimension of such a space is fundamental to knowing how many linearly independent solutions exist. This helps in predicting the behavior and form of general solutions for these equations.

Vector Space Dimension

The **dimension of a vector space** is the number of vectors in the basis of the space that are linearly independent. For the solution space of a homogeneous linear differential equation of order \(n\), the dimension is \(n\). This provides a measure of the space's complexity.

• Vector Basis: The basis vectors span the entire space, and their number is the dimension of the space.
• Linear Constraints: Any set of more than \(n\) vectors functions in an \(n\)-dimensional space will be linearly dependent.

Thus, understanding the concept of dimension helps you realize why a set of solutions larger than this number can't be independent. It is important for ensuring you have the right number of solutions to describe any other solution fully.